Diameter of colorings under Kempe changes

Bonamy, Marthe; Heinrich, Marc; Ito, Takehiro; Kobayashi, Yusuke; Mizuta, Haruka; Mühlenthaler, Moritz; Suzuki, Ai; Wasa, Kiyotaka

doi:10.1016/j.tcs.2020.05.033

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…In that same paper, Mohar also conjectured that the set of k-colorings of a k-regular graph forms a Kempe class, and this was settled in [4,1]. For further details and examples, we refer the reader to [9,13,3,2]. It is worth mentioning that the study of Kempe equivalence is also directly relevant in statistical physics [10,11].…”

Section: Introductionmentioning

confidence: 93%

Kempe equivalence of $4$-critical planar graphs

Feghali¹

2021

Preprint

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Answering a question of Mohar from 2007, we show that for every 4-critical planar graph, its set of 4-colorings is a Kempe class. IntroductionLet G be a graph, and let k be a non-negative integer.A Kempe chain in colors {a, b} is a maximal connected subgraph K of G such that every vertex of K has color a or b. By swapping the colors a and b on K, a new coloring is obtained. This operation is called a K-change. If c 2 is a k-coloring obtained from a k-coloring c 1 by a single K-change, then we write c 1 ∼ k c 2 . Two k-colorings c 1 and c 2 are K-equivalent (or K k -equivalent) if c 1 be obtained from c 2 by a sequence of K-changes.LetKempe chains were introduced by Kempe in his failed attempt at proving the Four Colour Theorem. Nevertheless, they have proved to be one of the most useful tools in graph coloring theory.Meyniel [8] showed that the set of 5-colorings of a planar graph forms a single Kempe class, and Las Vergnas and Meyniel [7] extended this result to K 5 -minor-free graphs. Mohar [9] showed that the set of 4-colorings of every 3-colorable planar graph is a Kempe class, and asked whether his result can be extended to 4-critical planar graphs (a graph G is 4-critical if it is not 3-colorable, but every proper subgraph of G is 3-colorable). In that same paper, Mohar also conjectured that the set of k-colorings of a k-regular graph forms a Kempe class, and this was settled in [4,1]. For further details and examples, we refer the reader to [9,13,3,2]. It is worth mentioning that the study of Kempe equivalence is also directly relevant in statistical physics [10,11].In this note, we answer Mohar's aforementioned problem in the positive.Theorem 1. Let G be a 4-critical planar graph. Then Kc(G, 4) = 1.

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Section: Introductionmentioning

confidence: 93%

Kempe equivalence of $4$-critical planar graphs

Feghali¹

2021

Preprint

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“…Conjecture 1. Let G be a planar graph, and L be a list assignment for G such that ≥   L v ( ) 5 for each ∈ v V G ( ). Then the set of L-colorings of G forms a Kempe class.…”

Section: Final Remarksmentioning

confidence: 99%

“…In that same paper, Mohar also conjectured that the set of

k

‐colorings of a

k

‐regular graph forms a Kempe class, and this was settled in [2, 9]. For further details and examples, we refer the reader to [4, 5, 7, 16, 22]. The edge‐coloring version has also been considered in several papers [1, 3, 13, 18] so has the vertex‐coloring version where only trivial Kempe changes are allowed [6, 8].…”

Section: Introductionmentioning

confidence: 99%

Kempe equivalence of 4‐critical planar graphs

Feghali

2022

Journal of Graph Theory

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“…The task is to decide if we can transform I s into I t by repeatedly applying a modification rule while maintaining the feasibility of intermediate solutions. Reconfiguration problems are studied for several problems such as Independent Set [17,27,8,12,13,18,4,10,28,3,25], Dominating Set [34,16,29], Clique [26], Matching [9,5,22], and Graph Coloring [7,31,6]. (See also surveys [35,30].…”

Section: Introductionmentioning

confidence: 99%

Independent set reconfiguration on directed graphs

Ito¹,

Iwamasa²,

Kobayashi³

et al. 2022

Preprint

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Directed Token Sliding asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. Directed Token Sliding is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions.In this paper, we initiate the algorithmic study of Directed Token Sliding. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a lineartime algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees [Demaine et al. TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which admits an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

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Diameter of colorings under Kempe changes

Cited by 5 publications

References 15 publications

Kempe equivalence of $4$-critical planar graphs

Kempe equivalence of $4$-critical planar graphs

Kempe equivalence of 4‐critical planar graphs

Independent set reconfiguration on directed graphs

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